Sunday, February 6, 2011

Shell's method to find volume

Definition: Shell method: Let R be a plane region bounded above by a continuous curve y = f(x), below by the x-axis, and on the left and right by x = a and x = b, then the volume of the solid of revolution obtained by rotating R about the y-axis is given by V:

Compute the volume of the solid generated by revolving the region bounded by y = x and y = x2 about each of the coordinate axes using the shell method.

We have y = x and y = x2. Thus, when using the shell method about the x-axis, we have x = y and x = y1/2. Thus, from wolfram alpha, we have

So what is the bottom line? If you haven't taken calculus in a while, I would say an important thing to remember would be do be clear on what axis you are revolving about. In our example, since we are revolving around the x axis, the washer method would probably be easier.
Here's a brief treatment: